Algebra II Formulas Flashcards
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Questions and Answers

What is the point slope form?

  • x² + bx + c = 0
  • x³ - y³ = (x - y)(x² + xy + y²)
  • y = mx + b
  • y - y1 = m(x - x1) (correct)

    • What is the slope intercept form?

  • y = mx + b (correct)
  • y - y1 = m(x - x1)
  • d = √((x2 - x1)² + (y2 - y1)²)
  • x³ + y³ = (x + y)(x² - xy + y²)

    • Define the difference of cubes.

    x³ - y³ = (x - y)(x² + xy + y²)

    Define the sum of cubes.

    <p>x³ + y³ = (x + y)(x² - xy + y²)</p> Signup and view all the answers

    What is the quadratic formula?

    <p>x = -b ± √(b² - 4ac) / 2a</p> Signup and view all the answers

    What is the standard form of a quadratic equation?

    <p>ax² + bx + c = 0</p> Signup and view all the answers

    What is the standard form for the equation of a circle?

    <p>(x - m)² + (y - n)² = r²</p> Signup and view all the answers

    Define the difference of squares.

    <p>x² - y² = (x + y)(x - y)</p> Signup and view all the answers

    What is the representation of imaginary numbers?

    <p>√(-1) = i</p> Signup and view all the answers

    Define the distance formula.

    <p>d = √((x2 - x1)² + (y2 - y1)²)</p> Signup and view all the answers

    What is the midpoint formula?

    <p>( (x1 + x2)/2 , (y1 + y2)/2 )</p> Signup and view all the answers

    What condition must hold for a graph to be symmetric with respect to the x-axis?

    <p>If (x,y) is on the graph, then (x,-y) is also on the graph.</p> Signup and view all the answers

    What condition must hold for a graph to be symmetric with respect to the y-axis?

    <p>If (x,y) is on the graph, then (-x,y) is also on the graph.</p> Signup and view all the answers

    What condition must hold for a graph to be symmetric with respect to the origin?

    <p>If (x,y) is on the graph, then (-x,-y) is also on the graph.</p> Signup and view all the answers

    Define completing the square.

    <p>x² + bx + (b/2)² = (x + b/2)²</p> Signup and view all the answers

    What is the formula for the slope of a line passing through two points?

    <p>m = (y2 - y1) / (x2 - x1)</p> Signup and view all the answers

    What condition indicates that two lines are parallel?

    <p>The slopes are equal, m1 = m2.</p> Signup and view all the answers

    What condition indicates that two lines are perpendicular?

    <p>The slopes are negative reciprocals, m1 = -1/m2.</p> Signup and view all the answers

    Define an even function.

    <p>f(-x) = f(x)</p> Signup and view all the answers

    Define an odd function.

    <p>f(-x) = -f(x)</p> Signup and view all the answers

    What is the greatest integer function?

    <p>Slanted ladder</p> Signup and view all the answers

    What is a vertical shift upward in terms of a function?

    <p>h(x) = f(x) + c</p> Signup and view all the answers

    What is a vertical shift downward in terms of a function?

    <p>h(x) = f(x) - c</p> Signup and view all the answers

    What is a horizontal shift to the right in terms of a function?

    <p>h(x) = f(x - c)</p> Signup and view all the answers

    What is a horizontal shift to the left in terms of a function?

    <p>h(x) = f(x + c)</p> Signup and view all the answers

    What represents a reflection in the x-axis?

    <p>h(x) = -f(x)</p> Signup and view all the answers

    What represents a reflection in the y-axis?

    <p>h(x) = f(-x)</p> Signup and view all the answers

    How is the sum of two functions represented?

    <p>(f + g)(x) = f(x) + g(x)</p> Signup and view all the answers

    How is the difference of two functions represented?

    <p>(f - g)(x) = f(x) - g(x)</p> Signup and view all the answers

    How is the product of two functions represented?

    <p>(fg)(x) = f(x)g(x)</p> Signup and view all the answers

    How is the quotient of two functions represented?

    <p>(f / g)(x) = f(x) / g(x)</p> Signup and view all the answers

    How is the composition of two functions represented?

    <p>(f o g)(x) = f(g(x))</p> Signup and view all the answers

    Study Notes

    Algebra II Formulas

    • Point Slope Form:
      Useful for writing the equation of a line given a point ((x_1, y_1)) and the slope (m). Formula: (y - y_1 = m(x - x_1)).

    • Slope Intercept Form:
      Represents linear equations, where (m) is the slope and (b) the y-intercept. Formula: (y = mx + b).

    • Difference of Cubes:
      A polynomial identity used to factor the difference of two cubes. Formula: (x^3 - y^3 = (x - y)(x^2 + xy + y^2)).

    • Sum of Cubes:
      A polynomial identity for factoring the sum of two cubes. Formula: (x^3 + y^3 = (x + y)(x^2 - xy + y^2)).

    • Quadratic Formula:
      Used to find the roots of a quadratic equation (ax^2 + bx + c = 0). Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}).

    • Quadratic Equation:
      A second-degree polynomial equation expressed as (ax^2 + bx + c = 0).

    • Standard Form for Equation of a Circle:
      Represents a circle's equation, where ((m,n)) is the center and (r) is the radius. Formula: ((x - m)^2 + (y - n)^2 = r^2).

    • Difference of Squares:
      A factoring identity for the difference of two squares. Formula: (x^2 - y^2 = (x + y)(x - y)).

    • Imaginary Numbers:
      Defined by the square root of negative one, represented as (i), where (i = \sqrt{-1}).

    • Sum of Squares:
      Representation involving squares of numbers, though it cannot be factored into real numbers. Formula: (x^2 + y^2 = (x^2 + xy + y^2)).

    • Distance Formula:
      Used to calculate the distance (d) between two points ((x_1, y_1)) and ((x_2, y_2)). Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).

    • Midpoint Formula:
      Determines the midpoint between two points. Formula: (\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)).

    • Symmetry in Graphs:

      • X-axis: A graph is symmetric with respect to the x-axis when ((x, y)) corresponds to ((x, -y)).
      • Y-axis: Symmetry around the y-axis occurs when ((x, y)) corresponds to ((-x, y)).
      • Origin: Symmetry about the origin is present when ((x, y)) corresponds to ((-x, -y)).

    • Completing the Square:
      A method to transform quadratic equations into vertex form. Formula: (x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2).

    • Principal Square Root of a Number:
      For a negative number (-a), the square root is defined in terms of imaginary numbers. Formula: (\sqrt{-a} = \sqrt{ai}).

    • Slope of a Line:
      The slope (m) of a line passing through two points can be calculated. Formula: (m = \frac{y_2 - y_1}{x_2 - x_1}).

    • Parallel Lines:
      Two lines are parallel if their slopes are equal, represented as (m_1 = m_2).

    • Perpendicular Lines:
      Two lines are perpendicular if their slopes are negative reciprocals, represented as (m_1 = -\frac{1}{m_2}).

    • Even Function:
      A function (f) is even if it satisfies (f(-x) = f(x)).

    • Odd Function:
      A function (f) is odd if it satisfies (f(-x) = -f(x)).

    • Greatest Integer Function:
      Often referred to as the "floor function," visualized like a slanted ladder.

    • Vertical Shifts:

      • Upward Shift: (h(x) = f(x) + c).
      • Downward Shift: (h(x) = f(x) - c).

    • Horizontal Shifts:

      • Right Shift: (h(x) = f(x - c)).
      • Left Shift: (h(x) = f(x + c)).

    • Reflections:

      • In the x-axis: (h(x) = -f(x)).
      • In the y-axis: (h(x) = f(-x)).

    • Operations on Functions:

      • Sum: ((f + g)(x) = f(x) + g(x)).
      • Difference: ((f - g)(x) = f(x) - g(x)).
      • Product: ((fg)(x) = f(x)g(x)).
      • Quotient: ((f / g)(x) = \frac{f(x)}{g(x)}).

    • Composition of Functions:
      Describes the output when a function (g) is applied to the input of another function (f). Formula: ((f \circ g)(x) = f(g(x))).

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    Test your knowledge of key Algebra II formulas with this set of flashcards. Each card presents a crucial mathematical concept, such as point-slope form and the quadratic formula, making it a perfect tool for review and practice. Strengthen your understanding of algebraic principles effectively!

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